Comparison principle without strict extrema for the infinity Laplace equation

Let $\Omega$ be the domain under consideration. Assume that $u,-v\in\operatorname{USC}(\bar\Omega)$ and $f\in C(\Omega)$ satisfy

$$-\Delta_\infty u\leq f\leq -\Delta_\infty v\quad\text{in }\Omega.$$

Suppose that either $u$ has no strict local maximum in $\Omega$ or $v$ has no strict local minimum in $\Omega$. No-extrema comparison conjecture. Then

$$\max_{\bar\Omega}(u-v)=\max_{\partial\Omega}(u-v).$$

This would yield a comparison principle under the stated one-sided absence of strict extrema and, in particular, would support uniqueness for the Dirichlet problem when the source term is nonnegative or nonpositive. The source presents it as a conjecture motivated by a preceding comparison theorem; its resolution is not established in the supplied text.